Specifically, \(A_k = \sum _{\mathcal {I}} \alpha ^{\mathcal {I}} {E}_{\mathcal {I}}\)  is called the multivector equivalent of \(\alpha _k\). This is a realization of the isomorphism between \({M}{}\)  and \(\Omega (M)\)  as \(C^\infty (M)\) -modules and it can be viewed as an extension of the musical isomorphisms between vectors and 1-forms [1]}. The multivector equivalent of the Riemannian volume form \(\mu \)  is \(^{-1^\dagger }\)  and \({I}(x)\)  represents the tangent space \(T_x M\) . Of course, in the case where \(g\)  is definite, the equivalent to \(\mu \)  is simply \(\) . On \(\partial M\) , we have the boundary pseuodoscalar \({I}_\partial \)  and dual to this the boundary normal vector field \(= _\partial ^\perp \) . As on \(M\) , the boundary pseudoscalar \({I}_\partial \)  is the multivector equivalent of the boundary area form \(\mu _\partial \) . From this point forward, we work solely with multivector fields and contract with directed measures to integrate. We now realize the action of \(\wedge \)  and \(\)  as the equivalents of \(d\)  and \(\delta \)  by

d k = (Ak) dXk+1,  k = (Ak)dXk-1.
